1. Field of the Invention
This application is related to the field of digital communications, and more particularly, to the field of secure digital communications.
2. Description of Related Art
Digital communications entails encoding a particular message onto carrier analog signals with different properties, and, at the receiving end, decoding the message based on the properties of the received signal.
Phase shift keying and frequency shift keying are digital signaling techniques discussed in R. N. McDonough & A. D. Whalen, Detection of Signals in Noise, 2nd ed., 1995. These techniques, however, can allow an eavesdropper to relatively easily decode the message. Spread spectrum techniques such as frequency hopping can be used to increase the privacy of the communications link. However, even these messages can be exposed using time-frequency signal analysis methods such as short time Fourier transform, Wigner-Ville distribution, and Wavelet transforms. Particle filtering methods described in N. D. Sidropoulos et al., “Tracking a frequency hopped signal using particle filtering”, ICASSP 2006 Proceedings, pp. 25-28 can be used to automatically track the frequency of a signal.
The relatively weak security of common methods can be attributed to the reliance of most modern communication protocols on the theory of linear vector spaces. For example, commonly used carrier analog signals s(t) can be analyzed using a linear decomposition
      s    ⁡          (      t      )        ≈            ∑              k        =        1            ∞        ⁢                  ⁢                  c        k            ⁢                        φ          k                ⁡                  (          t          )                    where Φ(t), k=1, . . . , ∞ form a complete representation of the space of finite energy functions. These functions can be, for example, simple sinusoids of differing frequency and phase, Gabor atoms, Wavelets, etc. Signal detection and demodulation can be done by computing the coefficients ck of the linear expansion and noting that these are different messages.
Several communications protocols use chaos. One of these protocols is described in S. Hayes et al., “Communication with chaos”, Physical Review Letters, vol. 70, pp. 3031-3034, 1993. This protocol depends on chaotic synchronization. Another protocol, described in Y. Hwang et al., “Physical-layer secrecy AWGN via a class of chaotic DS/SS systems: analysis and design”, IEEE Transactions on Signal Processing, Vol. 52, pp. 2637-2649, 2004, depends on the knowledge and ability to implement initial conditions exactly. These complications can make implementation difficult.